A large number of studies use polysomnography to continuously and non-invasively record electrophysiological signals during sleep with the goal of using these data to elucidate the biological pathways through which sleep affects health and functioning. Polysomnographic signals such as electroencenphograms and heart rate variability measure complex and dynamic processes whose frequency domain properties provided valuable and interpretable information. A dearth of tractable statistical models and methods for quantifying associations between frequency domain properties of collections of nonstationary time series with other study variables, such as clinical outcomes and experimental conditions, has limited the scope of questions that can be accurately addressed by analyzing polysomographic data. The goal of this research project is to develop a framework based on stochastic semiparametric evolutionary transfer functions for the spectral analysis of electrophysiological time series collected during sleep studies. Three specific aspects of this framework will be explored: (1) A time-varying spectral analogue of mixed effects models that will allow for the semiparametric analysis of the association between evolutionary power spectra and other variables while accounting for dependencies among correlated signals. (2) A procedure for discriminating between populations of nonstationary time series, such as between participants that respond positively to a treatment from those who do not. (3) A time-frequency canonical correlation analysis for obtaining low-dimensional measures of association between high-dimensional time series and processes measured by large collections of correlated variables. For each aspect of the framework, estimators will be developed, theoretical and empirical properties will be established, and efficient algorithms and computer programs will be created. These new methods will be used to analyze data from two existing studies: a clinical experimental study of sleep in older adults and a multi-cultural epidemiological study of sleep in women during the menopausal transition.